The cost of producing x units of a certain product is given by: C=10,000 + 5x+ 1/9 x^2
Find the value of x that gives the minimum average cost

You have a cost function for x units given by:
C = 10,000 + 5x + (1/9)x^2
Now, the cost per unit is going to be that same function divided by x.
Let's call this function U:
U = 10,000/x + 5 + (1/9)x
To find the minimum total cost, you need to find the minimum of
this function. Analyzing the derivatives should get you the answer
you need:
U' = -10,000/(x^2) + 1/9
U'' = 20,000/(x^3)
http://mathforum.org/library/drmath/view/53402.html
By examining U'= 0, we find that x = 300 is a critical point.
Further, since U'' is positive, the function must be at a minimum at
that point.